Invited Talk 6: A Century of the Tensor Network Formulation from the Ising Model

A hundred years have passed since Ising model was proposed by Lenz in 1920. One finds that the square lattice Ising model is already an example of two-dimensional tensor network (TN), which is formed by contracting 4-leg tensors. In 1941, Kramers and Wannier assumed a variational state in the form of the matrix product state (MPS), and they optimized it `numerically'. Baxter reached the concept of the corner-transfer matrix (CTM), and performed a variational computation in 1968. Independently from these statistical studies, MPS was introduced by Affleck, Lieb, Kennedy and Tasaki (AKLT) in 1987 for the study of one-dimensional quantum spin chain, by Derrida for asymetric exclusion processes, and also (implicitly) by the establishment of the density matrix renormalization group (DMRG) by White in 1992. After a brief (?) introduction of these prehistories, I'll speak about my contribution to this area, the applications of DMRG and CTMRG methods to two-dimensional statistical models, including those on hyperbolic lattices, fractal systems, and random spin models. Analysis of the spin-glass state, which is related to learning processes, from the view point of the entanglement structure would be a target of future studies in this direction.